| L(s) = 1 | − 2.41·3-s − 1.58·7-s + 2.82·9-s − 1.41·11-s + 0.171·13-s + 17-s + 19-s + 3.82·21-s − 9.24·23-s + 0.414·27-s − 5.82·29-s + 2.24·31-s + 3.41·33-s + 8.48·37-s − 0.414·39-s + 4.24·41-s − 10.2·43-s − 4.48·49-s − 2.41·51-s − 11.4·53-s − 2.41·57-s + 12.8·59-s + 5.75·61-s − 4.48·63-s − 13.2·67-s + 22.3·69-s + 10.5·71-s + ⋯ |
| L(s) = 1 | − 1.39·3-s − 0.599·7-s + 0.942·9-s − 0.426·11-s + 0.0475·13-s + 0.242·17-s + 0.229·19-s + 0.835·21-s − 1.92·23-s + 0.0797·27-s − 1.08·29-s + 0.402·31-s + 0.594·33-s + 1.39·37-s − 0.0663·39-s + 0.662·41-s − 1.56·43-s − 0.640·49-s − 0.338·51-s − 1.57·53-s − 0.319·57-s + 1.67·59-s + 0.737·61-s − 0.565·63-s − 1.61·67-s + 2.68·69-s + 1.25·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5004513787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5004513787\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.41T + 3T^{2} \) |
| 7 | \( 1 + 1.58T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 23 | \( 1 + 9.24T + 23T^{2} \) |
| 29 | \( 1 + 5.82T + 29T^{2} \) |
| 31 | \( 1 - 2.24T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 5.48T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80947523185228543754495769314, −7.02121660362294126842324014634, −6.25252444222672631133492188417, −5.90490562452928981178701799306, −5.22842547368811812480699610547, −4.44900296610593197984581244290, −3.67408930959227333450210990623, −2.69100625026331365960658791130, −1.58695534124159590387967947251, −0.38117839939102305150315056283,
0.38117839939102305150315056283, 1.58695534124159590387967947251, 2.69100625026331365960658791130, 3.67408930959227333450210990623, 4.44900296610593197984581244290, 5.22842547368811812480699610547, 5.90490562452928981178701799306, 6.25252444222672631133492188417, 7.02121660362294126842324014634, 7.80947523185228543754495769314
